Generating Functions for Moments of the Quasi–nilpotent Dt–operator

We prove a recursion formula for generating functions of certain renormalizations of ∗–moments of the DT(δ0, 1)–operator T , involving an operation ⊙ on formal power series and a transformation E that converts ⊙ to usual multiplication. This recursion formula is used to prove that all of these generating functions are rational functions, and to find a few of them explicitly. Introduction In combinatorics, one of the most useful methods for studying a sequence is to give its generating functions. The two most common types of generating functions are ordinary generating functions ∑ f(n)x, and exponential generating functions ∑ f(n)x/n!. In this paper we study the ∗–moment generating functions — a family of multivariable power series Fn— of a particular operator T that arose in the theory of free probability. We prove that Fn’s are all rational by applying a linear transformation between these two types of generating functions. The central object of this paper is the collection of ∗–moments of a particular bounded operator T on Hilbert space, which was constructed in [1] and which is a candidate for an operator without a nontrivial hyperinvariant subspace. (A hyperinvariant subspace of an operator T on a Hilbert space H is a closed subspace H0 ⊆ H that is invariant under every operator S that commutes with T , i.e. S(H0) ⊆ H0. It is an open problem whether every operator on Hilbert space that is not a multiple of the identity has a nontrivial hyperinvariant subspace.) The von Neumann algebra generated by T has a unique normalized trace τ , and by the ∗–moments of T we mean the values M(k1, l1, . . . , kn, ln) = τ ( (T )1T l1 . . . (T )nT ln ) , with n ∈ N, k1, . . . , kn, l1, . . . , ln ∈ N ∪ {0}. These ∗–moments determine a representation of T on a Hilbert space, (which can be shown to be bounded, see [1]), via the construction of Gelfand, Naimark and Segal, (cf. [2]) and hence they encode all essential properties of the operator. Our effort to understand the ∗–moments is part of an attempt better to understand the operator T . Date: Jan. 10, 2002. K.D. supported in part by NSF grant DMS–0070558. He thanks also the Mathematical Sciences Research Institute, where part of this work was carried out. Research at MSRI is supported in part by NSF grant DMS–9701755. C.Y. supported in part by NSF grant DMS-0070574. She is also partly supported by NSF grant DMS-9729992 through the Institute for Advanced Study. 1